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Implicitly restarted Arnoldi/Lanczos Methods for Large Scale Eigenvalue Calculations
, 1996
"... Eigenvalues and eigenfunctions of linear operators are important to many areas of applied mathematics. The ability to approximate these quantities numerically is becoming increasingly important in a wide variety of applications. This increasing demand has fueled interest in the development of new m ..."
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Cited by 20 (3 self)
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Eigenvalues and eigenfunctions of linear operators are important to many areas of applied mathematics. The ability to approximate these quantities numerically is becoming increasingly important in a wide variety of applications. This increasing demand has fueled interest in the development of new methods and software for the numerical solution of largescale algebraic eigenvalue problems. In turn, the existence of these new methods and software, along with the dramatically increased computational capabilities now available, has enabled the solution of problems that would not even have been posed five or ten years ago. Until very recently, software for largescale nonsymmetric problems was virtually nonexistent. Fortunately, the situation is improving rapidly. The purpose of this article is to provide an overview of the numerical solution of largescale algebraic eigenvalue problems. The focus will be on a class of methods called Krylov subspace projection methods. The wellknown Lanczos method is the premier member of this class. The Arnoldi method generalizes the Lanczos method to the nonsymmetric case. A recently developed variant of the Arnoldi/Lanczos scheme called the Implicitly Restarted Arnoldi Method is presented here in some depth. This method is highlighted because of its suitability as a basis for software development.
Accelerating The Lanczos Algorithm Via Polynomial Spectral Transformations
 Rice University
, 1997
"... . We consider the problem of computing a few clustered and/or interior eigenvalues of a symmetric matrix A without using a matrix factorization. This can be done by applying the Lanczos algorithm to p(A), where p() is a polynomial that maps the clustered and/or interior eigenvalues of A to extremal ..."
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Cited by 5 (1 self)
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. We consider the problem of computing a few clustered and/or interior eigenvalues of a symmetric matrix A without using a matrix factorization. This can be done by applying the Lanczos algorithm to p(A), where p() is a polynomial that maps the clustered and/or interior eigenvalues of A to extremal and well seperated eigenvalues of p(A). We will demonstrate several ways of constructing these polynomials by some standard approximation scheme such as minmax and least square. Key words. Lanczos method, eigenvalues, polynomial approximation AMS subject classifications. Primary 65F15, Secondary 65G05 1. Introduction. The symmetric eigenvalue problem Az = z (1.1) arises in many areas of science and engineering. We consider the problem of computing a few interior and/or clustered eigenvalues of a symmetric matrix A. This is often done by applying the Lanczos algorithm to the shifted and inverted operator (A \Gamma ¯I) \Gamma1 . Through the rational transformation /() = 1 \Gamma¯ , clus...
CRPC Research into Linear Algebra Software for High Performance Computers
, 1994
"... In this paper we look at a number of approaches being investigated in the Center for Research on Parallel Computation (CRPC) to develop linear algebra software for highperformance computers. These approaches are exemplified by the LAPACK, templates, and ARPACK projects. LAPACK is a software library ..."
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Cited by 4 (2 self)
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In this paper we look at a number of approaches being investigated in the Center for Research on Parallel Computation (CRPC) to develop linear algebra software for highperformance computers. These approaches are exemplified by the LAPACK, templates, and ARPACK projects. LAPACK is a software library for performing dense and banded linear algebra computations, and was designed to run efficiently on high performance computers. We focus on the design of the distributed memory version of LAPACK, and on an objectoriented interface to LAPACK. The templates project aims at making the task of developing sparse linear algebra software simpler and easier. Reusable software templates are provided that the user can then customize to modify and optimize a particular algorithm, and hence build a more complex applications. ARPACK is a software package for solving large scale eigenvalue problems, and is based on an implicitly restarted variant of the Arnoldi scheme. The paper focuses on issues impact...
Accelerating the Arnoldi Iteration  Theory and Practice
, 1998
"... The Arnoldi iteration is widely used to compute a few eigenvalues of a large sparse or structured matrix. However, the method may suffer from slow convergence when the desired eigenvalues are not dominant or well separated. A systematic approach is taken in this dissertation to address the issue of ..."
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Cited by 4 (2 self)
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The Arnoldi iteration is widely used to compute a few eigenvalues of a large sparse or structured matrix. However, the method may suffer from slow convergence when the desired eigenvalues are not dominant or well separated. A systematic approach is taken in this dissertation to address the issue of how to accelerate the convergence of the Arnoldi algorithm within a subspace of limited size. The acceleration strategies presented here are grouped into three categories. They are the method of restarting, the method of spectral transformation and the Newtonlike acceleration. Simply put, the method of restarting repeats a kstep Arnoldi iteration after improving the starting vector. The method is further divided into polynomial and rational restarting based on the way the starting vector is modified. We show that both mechanisms can be implemented in an implicit fashion by relating the restarted Arnoldi to a truncated QR...